Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ is coarsely equivalent to $E$. Let us also assume that $K$ is uniformly discrete, i.e. that the distance between distinct points of $K$ is bounded from below by a positive constant. Consider the Vietoris-Rips complex of $K$. It depends on a parameter delta (a positive number). What can one say about the homology of this complex for large delta? Is it true that it stabilizes for sufficiently large delta? Is the limit isomorphic to the homology of $E$ (that is, trivial in degree greater than $0$)? There is also a "Borel-Moore" version of this question, where one allows locally-finite but not necessarily finite chains.

  • $\begingroup$ "this means that $K$ is coarsely equivalent to $E$". No: this means that the embedding of $K$ into $E$ is a coarse equivalence. (It's maybe true that $E$ is not coarsely equivalent to any non-cobounded subset, but if so that's certainly not a definition, but a theorem.) $\endgroup$ – YCor Aug 15 '19 at 5:58

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